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I've been studying the R-formulae, and I am slightly stuck on the fact that adding two sinusoidal functions, like $\cos{x}+\sin{x}$ can form another sinusoidal function, namely $\sqrt{2}\sin({x+\frac{\pi}{4}})$.

It makes intuitive sense that something like $A\sin{x}+B\sin{x}$ form another sinusoidal wave, as they are 'in phase', as in, their periods line up, so they form a sinusoidal wave with the same period. However, it doesn't make much sense to me how waves whose periods do not line up still manage to form a smooth sinusoidal wave. As in; something like $\cos{x}+\sin{x}$ doesn't seem to have any meaning to me; how do 2 waves with mismatching periods form one consistent sine wave?

Are there any proofs/geometric intuitions that I should be thinking about? I've been messing around in GeoGebra and haven't found anything meaningful.

Habeeb M
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    Do you know identities like $\sin (x+\theta) = \sin x\cos \theta + \cos x \sin \theta$? – peterwhy Mar 02 '22 at 21:18
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    The general result is that (1) $\sin(x) = \cos(\pi/2 - x)$ (so all sinusoidal waves can be written as a sine), and (2) there are sum-to-product formulæ which allow one to combine sums into products. All of these can be proved. – Xander Henderson Mar 02 '22 at 21:19
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    For your particular example $\cos x + \sin x$, try to find an $r$ and $\theta$ so that $r\sin \theta = 1$ (the coefficient of $\cos x$) and $r\cos\theta = 1$ (the coefficient of $\sin x$), then

    $$\cos x + \sin x = r \sin \theta\cos x + r\cos \theta\sin x = r\sin (x+\theta)$$

     – peterwhy Mar 02 '22 at 21:22
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    While I have not yet found a good dupe target, there are already a number of related posts on [math.se]: https://math.stackexchange.com/q/754049/, https://math.stackexchange.com/q/1315199/, https://math.stackexchange.com/q/2849945/ . Note that things generally don't work well if the frequencies are distinct. – Xander Henderson Mar 02 '22 at 21:25

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Excuse the messy sketch, but I think it should provide simple 'geometric' intuition. enter image description here

Penguino
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  • That actually makes quite a lot of sense, thanks! I gotta mess with this in geogebra, never thought about trying something like that. – Habeeb M Mar 02 '22 at 22:57
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    Yes - something like geogebra will certainly draw a better diagram than mine :). – Penguino Mar 02 '22 at 22:59